This pattern search activity will improve students' understanding of the commutative property and associative property.
Main Curriculum Tie:
Mathematics Grade 3
Understand properties of multiplication and the relationship between multiplication and division. 5.
Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
- unifix cubes (shared between 5-6 participants)
- centimeter paper and overhead
- blackline master of array of triangles front/back & overhead
- Grapes of Math by Greg Tang
- transparency copy of p. 7 of Grapes of Math or plastic ants
Math For all Seasons by Greg Tang
Navigating Through Algebra in Grades 3-5 (I Spy) Navigation Series
Background For Teachers:
Students have been working with mathematical properties since kindergarten
even though they haven’t been calling them by the formal names. You will
be exploring some of these properties in this activity. Below you will see the
name of the properties explored in third grade and a sample of each.
Commutative Property of Addition (sometimes called the Order Property of Addition)
4 + 5 = 5 + 4
*Commutative Property of Multiplication (sometimes called the Order Property
of Multiplication) 4 X 5 = 5 X 4
Associative Property of Addition (sometimes considered grouping) (6 + 4) +
2 = 6 + (4 + 2)
*Identity Property of Addition 3 + 0 = 3
*Identity Property of Multiplication 3 X 1 = 3
Zero Property of Multiplication 3 X 0 = 0
* Properties that are introduced in the third grade core.
Intended Learning Outcomes:
1. Demonstrate a positive learning attitude towards mathematics.
3. Reason mathematically.
4. Communicate mathematically.
5. Make mathematical connections.
6. Represent mathematical situations.
Invitation to Learn
Place transparency of p. 7 (ants) from Grapes of Math on the overhead or recreate
it using plastic ants. Read “Ant Attack” from p. 8 and encourage
participants to quickly count the ants before they are removed. Talk about their
strategies. Write the numeric expression that helped them count the ants.
- You may want to select a few properties to review such as the Commutative
and Associative. You may wish to review just the properties that have been
learned during the multiplication unit. Make the lesson fit your needs at
- Distribute copies of the blackline master showing an array of triangles
to the students. Ask how many triangles are in the array. Point out that there
are many interesting ways to find the answer. The object of the rest of this
activity will be not to count the individual 32 triangles, but to look for
various patterns in the array and to translate the visual patterns into numeric
equations. “In what ways will the Commutative, Associative, Identity,
or Zero Properties help you when looking for patterns?”
- Place a transparency copy of the blackline master on the overhead and discuss
possible numeric expressions. (The participants might start with an with a
simple one such as 1 + 3 + 5 + 7 + 7 + 5 + 3 + 1.) Did anyone look for “doubles”?
Did anyone “make tens”?
- Can you find a 5 X 4 rectangle inside this array of triangles? How would
you write the remaining numeric expression? [(5 X 4) + 4 + 4 + 2 + 2 or (5
X 4) + (2 X 4) + (2 X 2)]
Can demonstrate commutative property 20 + 4 + 4 + 2 + 2 = 2 + 2 + 4 + 4 +
20. Can demonstrate associative property (20 + 4 + 4) + 2 + 2 = 20 + (4 +
4 + 2 + 2).
- Explore and record as many patterns as possible with your partner.
Record the numeric expression below the picture after it is partitioned.
Think of another way to write the numeric expression and record it.
Be able to defend your reasoning as to why the different numeric expressions
have the same sum. Find ways of shortening numeric expressions.
F. Y. I. As I tried this activity with several classes, I found that the students
wanted their own activity sheet. The resource students struggle with writing
the numerical equation and may need extra help.
Math/Science—Create your own simple array and exchange with a
partner. How many different ways can the simple array be partitioned? Can you
record the different numerical equations?
Finish reading The Grapes of Math to the class. As a class, create a rhyming
math puzzle and picture similar to the book. Talk about how it is created and
how to find rhyming words. How would you give the clue in the last stanza? Ask
students to work in groups to create a “group” rhyming math puzzle
similar to the samples in The Grapes of Math. The group must also create a picture
to support the puzzle. Combine the group puzzles to create a class book of “Mind-Stretching
Homework & Family Connections
Take a new blackline copy and the completed “Pattern Search” copy.
Have the student teach their parent three patterns and the correlating numeric
equations. Together, they can discover another way to write the numeric equations
or find another undiscovered pattern. The parent should write a comment to their
child on the homework.
See rubric sample.
Created Date :
Aug 13 2003 16:49 PM